I wanted to share this sum with you all, though I have no clue how to prove it ---

\[\sum_{n=1}^{\infty}\left[(-1)^n \dfrac{(n!)^2}{n^2(2n)!}\right]=2\log(\phi)\log\left(\dfrac{1}{\phi}\right)\] where \(\phi\) is the golden ratio.

It has an amazing closed form, doesn't it?

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TopNewesthttp://mathworld.wolfram.com/CentralBinomialCoefficient.HTML gives a bit of info – Bogdan Simeonov · 2 years, 9 months ago

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click me – Bogdan Simeonov · 2 years, 9 months ago

Here's a proof for a lot of these( including this one ):Log in to reply